Interpolating Feigin-Frenkel Duality at the Critical Level to Matrices of Complex Size

Abstract: In this paper, we extend Feigin-Frenkel duality at the critical level to the setting of complex rank. This is accomplished by considering the center of a vertex algebra in Deligne's interpolating categories, along with Feigin's Lie algebras of complex rank, $\mathfrak{gl}{\lambda}$ and $\mathfrak{po}{\lambda}$. More precisely, we define the universal affine vertex algebras associated with Lie algebras in $\underline{\mathrm{Re}}\mathrm{p}(\mathrm{GL}{\alpha},\mathbb{F})$, $\underline{\mathrm{Re}}\mathrm{p}(\mathrm{O}{\alpha},\mathbb{F})$ and $\underline{\mathrm{Re}}\mathrm{p}(\mathrm{Sp}{\alpha},\mathbb{F})$, and describe their centers at the critical level explicitly by interpolating Molev's construction of Segal-Sugawara vectors. Using the formalism of Poisson vertex algebras, we identify a natural set of generators for the Drinfeld-Sokolov reduction of $\mathfrak{gl}{\lambda}$ and $\mathfrak{po}{\lambda}$, denoted by $\mathcal{W}(\mathfrak{gl}{\lambda})$ and $\mathcal{W}(\mathfrak{po}_{\lambda})$, respectively. Finally, we show that the interpolated Feigin-Frenkel isomorphism maps the interpolated Segal-Sugawara vectors to these generators.